Greece is looking to sell assets including utility companies, Athens' international airport and two of its main ports. The country is also looking for investments in several real estate developments which could interest the prince, who is the part owner of luxury hotels in London, New York and Paris.

Greece has been accused of dragging its feet on privatization. A government official who declined to be named played down Alwaleed's visit, however, telling Reuters: "It was a customary visit, there is interest in investing in Greece."

The move by Athens to look elsewhere for funding comes at a time when euro zone policymakers have been engaged in a public debate over whether the country needs a third bailout. International lenders have already granted Greece two bailouts worth around 240 billion euros but Greece faces a further funding gap of up to 10 billion euros ($13.1 billion ) by September, according to some accounts.

"There are two issues in Greece," Thanos Vamvakidis, head G10 FX Strategy at Bank of America Merrill Lynch Global Research told CNBC. "There is a funding gap in 2014 and 2015 and this needs to be covered, though it will be discussed in the October review (of the bailout program)."

Vamvakidis told CNBC Europe's "Squawk Box" that Greece needed a debt writedown of at least 10 percent of its gross domestic product (GDP). "It has to be done because most of the Greek debt is going to the rest of Europe, if they don't provide them with debt relief then Greece won't be able to repay them."

"The important thing is that they need to provide more money for Greece, the problem is with the design of the [bailout] program…[the program] assumed that Greece would reach a primary surplus of 4.5 percent and keep this for 20 years- no country has ever done this, it's impossible."